Since our proposed ODS-based QP, which computes control actions using the ILOS-based guidance law, has constraints on the
magnitude of the control input vector, it belongs to the family of trust-region subproblems (TRS), i.e., QPs over spheres and ellipsoids [1]}, [2]}. TRS has long been of interest to the optimization research community (see, e.g., the classical work by Forsythe and Golub [3]}); because a TRS needs to be solved in each step of trust-region optimization algorithms.
In this paper, we provide the necessary and sufficient conditions for the existence of solutions to the special TRS that arises
in the context of the magnetic microswimmer control problem. Furthermore, we provide sufficient conditions under which the absolute continuity of the generated control input is guaranteed. The absolute continuity of the control input is not only appealing from an existence and uniqueness of solutions perspective, but is also significant from a practical point of view. Indeed, it has been observed in practice that as long as a helical microswimmer is commanded a smooth desired-velocity profile with limits on acceleration, the highly correlated requirements of rotating below the step-out frequency as well as alignment of the field rotation axis with
the microrobot axis will be met [4]}.
